Brownian Motion in AdS/CFT - ggcyzha.github.io
Transcript of Brownian Motion in AdS/CFT - ggcyzha.github.io
Jilin University (via Zoom)
Masaki Shigemori
February 4, 2021
Based on work with Tatsuki Nakajima (Nagoya) and Hirano Shinji (Wits), arxiv:2012.03972
What is ๐๐?
2
๐ช๐๐ โก ๐๐ โ ฮ2
Various other expressions:
๐ = ๐ , ๐ = ๐ , ฮ = ๐
๐ช๐๐ =18(๐ ๐ โ ๐ ๐ )
= โ14det ๐ = โ
18๐ ๐ ๐ ๐
In 2D QFT, the ๐๐ operator is defined by
๐ง = ๐ฅ1 + ๐๐ฅ2
What is ๐๐?
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Nice properties:[Zamolodchikov 2004]
limโ
๐ ๐ง ๐ ๐ง โ ฮ ๐ง ฮ z = ๐ช๐๐(๐ง ) + (derivatives)
๐ช๐๐ = โจ๐โฉโจ๐โฉ โ โจฮโฉ2 ร Not just for |0โฉ but for |๐โฉ
๐๐ deformation
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` Non-renormalizable ( ๐ = massโ2 = length2).
` Still, surprisingly predictable
` Energy spectrum of a theory on a circle of radius ๐
๐ธ ๐ , ๐ = ๐ 1 โ 1 โ 2๐ถ๐ ๐ , ๐ถ = + โ๐/12
๐
` Integrable
` Thermodynamics (๐ < 0: Hagedorn)
` Deformed theory with (๐, ๐) โผ undeformed theory with (๐ , ๐ )
[Smirnov-Zamolodchikov โ16][Cavaglia, Negro, Szecsenyi, Tateo โ16]
๐de ormed = ๐ + ๐โซ ๐ช๐๐ (roughly)
Cf. [Haruna, Ishii, Kawai, Sakai, Yoshida โ20]
non-unitary
Holography
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` ๐๐: irrelevant in IR, relevant in UV
ร Non-normalizable in AdS
ร Change AdS asymptotics?
๐ = 0
AdS AdS
?
โผ ๐
โUndo decouplingโ?
deformBoundary
๐ ๐
Holography: moving field theory into bulk
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The deformed theory corresponds to placingthe field theory at ๐๐ โผ ๐ of the bulk(๐: radial coordinate in Fefferman-Graham coordinates)
[McGough, Mezei, Verlinde โ16][Guica-Monten โ19][Caputa-Datta-Jiang-Kraus โ20]
` ๐๐ deformation makes the bndy cond be naturally given at ๐๐
` CFT living at ๐ = 0 is โequivalentโ to deformed theory at ๐๐ โผ ๐
๐ = ๐de ormed + ๐ann lar
` Valid only in pure gravity without matter (so far)
๐ = 0 ๐
CFT lives here
Deformed theory
lives here
๐๐~๐
โannular regionโ
๐๐ and width of particles [Cardy-Doyon โ20] [Jiang โ20]
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Deformation makes particles have โwidthโ ๐
` ๐๐ deformation dynamically changes the metric
` ๐๐ deforming free scalars ร Nambu-Goto with tension โ๐[Cavaglia, Negro, Szecsenyi, Tateo โ16]
A lot more to explore about ๐๐ deformation!
โRandom geometryโ by Cardy
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Idea: rewrite ๐๐ using Hubbard-Stratonovich transformation
[Cardy โ18]
exp โ๐โซ ๐2 โผ โซ [๐โ] exp 1 โซ โ2 โ โซ โ๐
This talk:` Apply this to compute ๐๐-deformed ๐ correlators
` Previous results: 2pt func at ๐ช(๐2), 3pt func at ๐ช(๐)
` Develop a new method to compute ๐-correlators
` We computed 4pt func at ๐ช(๐)
` Result is applicable to any deformed CFT
deformation of backgnd geometry
(because ๐ โผ ๐๐
)
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What we do:
` Compute ๐ช ๐ correction to Liouville-Polyakov anomaly action
โก ๐ =0[๐] โ ๐ [๐]
` Vary backgnd metric ๐ โ ๐ + โ to compute ๐-correlators (explicit computations at ๐ช(๐))
Plan
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1. Introduction
2. ๐๐ deformation as random geometries
3. ๐๐-deformed Liouville action
4. Stress tensor correlators (technical!)
5. ๐๐-deformed OPEs
6. Discussions
3
๐๐ deformation
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` ๐๐-deformed theory is defined incrementally by:
๐ : stress-energy tensor of the ๐-deformed theory
๐ ๐ + ๐ฟ๐ โ ๐ ๐ =๐ฟ๐๐2
๐2๐ฅ ๐ ๐ช๐๐ โก ๐ฟ๐
* Our convention for ๐ :
` ๐๐-deformed theory with finite ๐ is obtained by iteration
` We will often suppress ๐2๐ฅ ๐
๐ฟ๐๐ =14๐
๐ ๐ฟ๐
๐๐ = random geometries
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` HS transformation:
๐โ ๐ = ๐8 โซ ๐ ๐ ๐ ๐
Saddle point: โโ = โ ๐ ๐ ๐ = ๐ช(๐ฟ๐)
Change in backgnd metric๐ โ ๐ + โ
โMaster formulaโ
Gaussian integral over โ
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๐๐-deformed theory with ๐, ๐
Undeformed theory with ๐ , ๐โฒ=
(up to the HS action)
๐
๐, ๐
deformedCFT
Equivalent theories
In saddle-pt approximation,
Cf. bulk picture
Parametrizing โ
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โ = โ ๐ผ + โ ๐ผ + 2๐ ฮฆ
In 2D, any โ = diff + Weyl
๐ฅ โ ๐ฅ + ๐ผ ๐๐ 2 โ ๐2ฮฆ๐๐ 2
Useful to write ฮฆ = ๐ โ 12โ ๐ผ
โMaster formulaโ
Summary so far
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โ = โ ๐ผ + โ ๐ผ + 2๐ ๐ โ 12โ โ ๐ผ
` ๐๐ = random geometries
` โMaster formulaโ
Liouville-Polyakov action (1)
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` Dependence of CFT2 on backgnd metric ๐ (๐ฅ) is determined by conformal anomaly [Polyakov โ81]:
= 1 for flat โ2
Cf. For general fiducial metric ๐,
Conformal gauge ๐๐ 2 = ๐2ฮฉ๐๐ง ๐ ๐ง
๐0 ๐2ฮฉ๐ = โ24 โซ๐2๐ฅ ๐ ๐ ๐ ฮฉ๐ ฮฉ + ๐ ฮฉ + ๐0 ๐
But note that we are not doing Liouville field theory
Liouville (-Polyakov) action
conformal-gauge Liouville action
Liouville-Polyakov action (2)
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` Valid for any CFT
` We can compute correlators โจ๐๐โฆ โฉ by varying ๐ โ ๐ + โ and taking derivatives of ๐0 with respect to โ
` The conformal form ๐0[๐2ฮฉ๐ฟ] contains the same information as ๐0[๐]
ร Can also use ๐0 ๐2 ๐ฟ to compute โจ๐๐โฆ โฉ
(We will come back to this point later)
๐๐-deforming Liouville action (1)
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` We could expand ๐0[๐ + โ] in โ
` But we take a different approach
๐โ ๐ [๐]+ ๐[๐] โผ ๐๐ผ ๐๐ ๐โ1
8 โซ โโโ๐ [๐+โ]
` Letโs consider how Liouville action ๐0[๐] is ๐๐-deformed (at ๐ช(๐ฟ๐))
From the deformed action ๐ฟ๐ ๐ , we can compute any โจ๐๐โฆ โฉfor any ๐๐-deformed CFT
Possible approaches:
Need to evaluate ๐0[๐ + โ]
๐๐-deforming Liouville action (2)
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` In 2D, we can bring ๐ + โ back to original metric via diff, up to Weyl rescaling
๐ (๐ฅ) + โ (๐ฅ) ๐๐ฅ ๐๐ฅ = ๐2ฮจ ๐ ๐ฅ ๐๐ฅ ๐๐ฅ
๐ฅ = ๐ฅ + ๐ด (๐ฅ) For some ๐ด ๐ฅ ,ฮจ(๐ฅ)
This is possible even for finite โ = โ ฮฑ + โ ฮฑ + 2๐ ฮฆ
ร easy to evaluateร ๐0 ๐(๐ฅ) + โ(๐ฅ) = ๐0[๐2ฮจ ๐ ๐ฅ ]
๐ด 2 ,ฮจ(2), โฆ are complicated (nonlocal)
Our approach:
๐๐-deforming Liouville action (3)
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For ๐2ฮจ ๐ ๐ฅ โก ๐ ๐ฅ ,
ร Using the master formula and carrying outGaussian integration, we can get ๐ฟ๐
๐๐-deforming Liouville action (4)
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๐ฟ๐[๐] = ๐ฟ๐saddle[๐] + ๐ฟ๐ l t[๐]
` Exact at ๐ช ๐ฟ๐
` Nonlocal (expected of ๐๐-deformed theory, but so was original LP action...)
` ๐ฟ๐ l t is fluctuation term coming from Gaussian integral.
` Contains contribution from ฮจ(2). Very complicated and divergent. Depends on measure.
` Vanishes after regularization (this can be shown using conformal pert theory)
` Non-vanishing at ๐ช(๐ฟ๐2) [Kraus-Liu-Marolf โ18]
` Can be dropped at large ๐ (๐ฟ๐saddle โผ ๐ +1๐ฟ๐ ), which is relevant for holography
๐๐-deforming Liouville action (5)
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๐ฟ๐ is very simple in conformal gauge:
` โLocalโ in ฮฉ but itโs really nonlocal (just like the CFT Liouville action)
` We will use this to compute ๐ correlators
Higher order
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` The same procedure gives a differential equation to determine the effective action ๐ [๐2ฮฉ๐ฟ] at finite deformation ๐:
` We ignored fluctuation (i.e., itโs valid at large ๐)
` Recursive relation:
where ๐ = โ!๐
` No longer โlocalโ in ฮฉ at ๐ช(๐2)
Summary so far
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Polyakov-Liouville action
Deformed action at ๐ช ๐ฟ๐
Was useful: โ = โ ฮฑ + โ ฮฑ + 2๐ ฮฆ
diff Weyl
Getting ๐ correlators (1)
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` We can get โจ๐๐โฆ โฉ by varying ๐ โ ๐ + โ,expanding ๐ ๐ + โ = ๐โ๐ [๐+โ] in โ,and reading off coefficients.
We know this.CFT: Liouville action ๐0deformed: ๐ฟ๐ we just computed
Getting ๐ correlators (2)
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` Flat background: ๐ = ๐ฟ, ๐๐ 2 = ๐๐ง๐ ๐ง.
Coeff of ๐๐ผ
Coeff of ๏ฟฝ๏ฟฝ๐ผ
Coeff of ๐
๐ = ๐
๐ = ๐
ฮ = ๐
e.g.
โ ๐๐ โผ 1โ
๐๐ ๐ฟ + โ โ โฌ โ
โ = ๐ ฮฑ + ๐ ฮฑ + 2๐ ฮฆ
Using conformal-gauge action
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` Can use conformal-gauge action ๐e ๐2ฮฉ๐ฟ instead of ๐e [๐] to compute ๐ correlators
1. Start with flat backgnd, ๐๐ 2 = ๐๐ง ๐ ๐ง
2. Vary ๐ฟ โ ๐ฟ + โ. ๐๐ 2 = ๐๐ง ๐ ๐ง + 2 ๐๐ผ ๐๐ง2 + ๏ฟฝ๏ฟฝ๐ผ ๐ ๐ง2 + ๐ ๐๐ง ๐ ๐ง .(Here ๐ผ, ๐ are finite.)
3. Find diff ๐ฅ โ ๐ฅ = ๐ฅ + ๐ด(๐ฅ) that brings metric into conformal gauge:
๐๐ 2 = ๐2ฮจ( , )๐๏ฟฝ๏ฟฝ ๐ ๏ฟฝ๏ฟฝ
4. Compute ๐e ๐2ฮจ๐ฟ = ๐e ๐ฟ + โ
5. Read off correlator from expansion
Similar to what we did when we computed deformed Liouville action. But here we need ๐ด,ฮจ to higher order to compute higher correlator ๐๐๐โฆ .
Check: the case of CFT
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To check that this method works, letโs apply it to some CFT correlators.
Conformal-gauge action:
๐e = ๐0 ๐2ฮฉ๐ฟ = โ๐
12๐๐2๐ง ๐ฮฉ ๏ฟฝ๏ฟฝฮฉ
Letโs reproduce the known expressions
๐1๐2 =๐
2๐ง124
๐1๐2๐3 =๐
๐ง122 ๐ง132 ๐ง232
2pt:
3pt:
CFT 2pt func (1)
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To see how it goes, letโs carry out this procedure for CFT.
๏ฟฝ๏ฟฝ = ๐ง + ๐ด(1) + ๐ด 2 + โฏ ,
Varied action:
๐0 ๐2ฮจ( , )๐ฟ = โ๐
12๐๐2๏ฟฝ๏ฟฝ ๐ ฮจ ๐zฮจ
๐ด(1) = ๐ผ
ฮจ = ฮจ 1 + ฮจ 2 +โฏ , ฮจ 1 = ฮฆ = ๐ โ (๐๐ผ + ๏ฟฝ๏ฟฝ๐ผ)
Want โจ๐๐โฉร Extract coeff of ๏ฟฝ๏ฟฝ๐ผ ๏ฟฝ๏ฟฝ๐ผ
= โ๐
12๐๐2๐ง ๐ ๐ โ ๐๐ผ + ๏ฟฝ๏ฟฝ๐ผ ๏ฟฝ๏ฟฝ ๐ โ ๐๐ผ + ๏ฟฝ๏ฟฝ๐ผ + (higher)
= โ๐
12๐๐2๐ง ๐ฮจ ๏ฟฝ๏ฟฝฮจ + (higher)
CFT 2pt func (2)
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๐0 ๐2ฮจ( , )๐ฟ โ โ๐
12๐ ๐2๐ง ๐2๐ผ ๐๏ฟฝ๏ฟฝ๐ผ
๏ฟฝ๏ฟฝ1๐ง= 2๐๐ฟ2(๐ง)
=๐
12๐ ๐2๐ง ๐3๐ผ ๏ฟฝ๏ฟฝ๐ผ
=๐
12๐ ๐2๐ง ๐31๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๐ผ ๏ฟฝ๏ฟฝ๐ผ
1๏ฟฝ๏ฟฝ=
12๐
๐2๐ง๐ง โ ๐งโฒ ๐3
1๏ฟฝ๏ฟฝ=โ3๐
๐2๐ง๐ง โ ๐ง 4
(above) =โ๐4๐ ๐2๐ง
๏ฟฝ๏ฟฝ๐ผ ๐ง ๏ฟฝ๏ฟฝ๐ผ ๐ง๐ง โ ๐ง 4
โจ๐๐ โฉ =๐
2 ๐ง โ ๐ง 4
Here
Therefore
3
Also,Cf. ฮ = โ 1
48๐
(โcreatedโ ๏ฟฝ๏ฟฝ๐ผ)
CFT 3pt func
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Conformal-gauge action ๐0[๐2ฮฉ๐ฟ] is quadratic. How can we get โจ๐๐๐โฉ?
1. Need to rewrite ๏ฟฝ๏ฟฝ, ๏ฟฝ๏ฟฝ in ๐0[๐2ฮจ , ๐ฟ] in terms of ๐ง, ๐ง
2. ฮจ = ฮจ 1 + ฮจ 2 +โฏ
๐0 ๐2ฮจ( , )๐ฟ = โ ๐12 โซ ๐2๏ฟฝ๏ฟฝ ๐ ฮจ(๐ฅ) ๐ ฮจ(๐ฅ)
๐ช(โ) ๐ช(โ2)
๐ ๏ฟฝ๏ฟฝ, ๏ฟฝ๏ฟฝ๐ ๐ง, ๐ง
๐2๐ง๐๐ง๐๏ฟฝ๏ฟฝ ๐ +
๐ ๐ง๐๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ
๐๐ง๐ ๏ฟฝ๏ฟฝ
๐ +๐ ๐ง๐ ๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ
๐1๐2๐3 = ๐` By similar manipulations, can check
๐ฅ + ๐ด(๐ฅ)
` All contributions are needed to reproduce the correct result
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Explicit forms of second-order terms:
Deformed ๐-correlators (1)
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We just reproduced CFT correlators.
Now consider deformed ones.
๐ฟ๐ ๐2ฮฉ๐ฟ =
= โ2(๐ฮฉ)(๏ฟฝ๏ฟฝฮฉ)(๐๏ฟฝ๏ฟฝฮฉ) + ๐ช(ฮฉ4)
Whatโs known in the literature at ๐ช(๐ฟ๐):` 3pt functions (based on conformal pert theory [Kraus-Liu-Marolf โ18],
and random geom and WT id [Aharony-Vaknin โ18])
` No 4pt functions
Deformed 3pt functions
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Deformed 3pt func is just like CFT 2pt func;Simply set ฮฉ โ ฮจ โผ ฮฆ = ๐ โ (๐๐ผ + ๏ฟฝ๏ฟฝ๐ผ) and also ๐ฅ โผ ๐ฅ
๐ฟ๐ ๐2ฮฉ๐ฟ โ
Straightforward to read off
Reproduces known results
Deformed 4pt functions
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Deformed 4pt func is similar to CFT 3pt func.
We have to take into account ๐ฅ = ๐ฅ + ๐ด(๐ฅ) and correction ฮจ 2 .Also, ๐ฟ๐ has quartic terms ๐ช(ฮฉ4) as well.
All contributions are needed for final results
Deformed OPEs?
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Can understand these from deformed OPEs?
OPEs from 3pt funcs (1)
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` 3pt func is reproduced
` Consistent with the flow equation ฮ = โ ๐๐
OPEs from 3pt funcs (2)
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More interesting:
= โ2๐ ๐ฟ๐๐
1๐ง โ ๐ค
๐๐งโฒ ๐(๐งโฒ) + (๐ง โ ๐ค)
` Can be regarded as coming from field-dependent diff[Conti, Negro, Tateo โ18,โ19] [Cardy โ19]
๐ ๐ง โ ๐ ๐ง + ๐ = ๐ + ๐๐ ๐, ๐ =๐ฟ๐๐ ๐๐ง ๐ ๐ง
๐ ๐ง ๐ ๐ค โ ๐ ๐ง ๏ฟฝ๏ฟฝ๐ ๐ค ๐ โผ โ 2๐โ
๐
` Non-local OPE
OPEs from 3pt funcs (3)
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Different pair in the same 3pt func:
` Again, comes from field-dependent diff
๐ ๐ง โ ๐ ๐ง + ๐ = ๐ + ๐๐ ๐, ๐ =๐ฟ๐๐ ๐๐ง ๐ ๐ง
๐ ๐ง ๐ ๐ค โ ๐ ๐ง ๐ ๏ฟฝ๏ฟฝ๐ ๐ค โผ ๐ ๐ง โซ ๐๐ง ๐ ๐ง ๏ฟฝ๏ฟฝ๐ ๐ค โ (above)
Consistency check with 4pt funcs
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Check OPEs derived above using 4pt funcs:
โ2๐ ๐ฟ๐๐ ๐ง34
๐๐ง โจ๐ ๐ง1 ๐(๐ง2)๐ ๐ง โฉ + (๐ง3 โ ๐ง4)
ร ๐๐ OPE at ๐ช(๐ฟ๐0)
ร ๐๐ OPE at ๐ช(๐ฟ๐)
Agrees with the ๐ง34 โ 0 expansion of the 4pt func
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Summary:` Studied the stress-energy sector of ๐๐-deformed theories using random
geometry approach
` Found ๐๐-deformed Liouville-Polyakov action exactly at ๐ช(๐ฟ๐)
` Derived equation to determine deformed action for finite ๐
` Developed technique to compute ๐-correlators
Correlators at ๐ช(๐ฟ๐) is computable also by conformal perturbation theory. Random geometry approach seems to allow straightforward generalization to higher order, at least formally
` Deformed OPEs show sign of non-locality
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Future directions:` Go to higher order in ๐ฟ๐
` Solve the differential equation for ๐ [๐]? Large ๐ (dual to classical gravity)?
` All-order correlators, such as ๐บฮ ๐ง12 โก โจฮ(๐ง1)ฮ(๐ง2)โฉ ?Expectation: ๐บฮ < 0 at short distances (negative norm)cf. [Haruna-Ishii-Kawai-Skai-Yoshida โ20]
` Better understand the fluctuation part ๐ l t
` Why does it vanish at ๐ช ๐ฟ๐ , as indicated by conf pert theory?
` Use vanishing of it at ๐ช ๐ฟ๐ to regularize ๐ l t
` More
` Compute physically interesting quantities
` Inclusion of matter ร modify the ๐๐ operator?
` Position-dependent coupling cf. [Chandra et al., 2101.01185]